A root loci plot is simply a plot of the s zero values and the s poles on a
graph with real and imaginary coordinates. The root locus is a curve of the location of the poles of a
transfer function as some parameter (generally the gain K) is varied. The number of zeros does not
exceed the number of poles.

The locus of the roots of the characteristic equation of the closed loop system
as the gain varies from zero to infinity gives the name of the method.
Such a plot shows clearly the contribution of each open loop pole or zero to the
locations of the closed loop poles. This method is very powerful graphical technique
for investigating the effects of the variation of a system parameter on the
locations of the closed loop poles. General rules for constructing
the root locus exist and if the designer follows them, sketching of the root loci
becomes a simple matter.

The closed loop poles are the roots of the characteristic
equation of the system. From the design viewpoint, in some
systems simple gain adjustment can move the closed loop poles to the desired
locations. Root loci are completed to select the best parameter value
for stability. A normal interpretation of improving stability is when
the real part of a pole is further left of the imaginary axis.

The open-loop transfer function between the forcing input R(s) and the measured output Y1(s) =

T1(s) = K.G(s)H(s)

Let G(s) = Q(s)/ P(s) and H(s) = W(s) /V(s) that is G(s)H(s) = Q(s).W(s) / P(s).V(s)
Then the open loop poles are the roots of the characteristic equation for the open loop transfer function.

P(s).V(s)= 0

The open loop zeros are roots of the W(s) Q(s)

W(s).Q(s)= 0

The closed-loop transfer function =

K is the value of the open loop gain . 1 + KG(s)H(s) is the characteristic equation

The closed loop poles ( when T(s) = infinity ) must satisfy

K.G(s).H(s) = -1

This is can be interpreted using vectors

Again letting G(s) = Q(s)/ P(s) and H(s) = W(s) /V(s) that is G(s)H(s) = Q(s).W(s) / P(s).V(s).
Then K(G(s).H(s) = -1 can be rewritten as K.Q(s).W(s) = - P(s)V(s) ....Therefore
The closed loop poles are roots of the characteristic equation for the closed loop system =

P(s).V(s) + K.Q(s)W(s) = 0

Generally the location of these roots in the s plane changes as the gain factor K is altered.
The root locus is the locus of these roots as a function of K.

For K = zero the roots of the above equation are roots of the P(s).V(s) which are the same as the poles of the
open loop transfer function G(s)H(s). If K becomes very large the roots approoach those of Q(s).W(s) which are the open-loops
zeros. Therefore as K is increased for zero to infinity the loci of the closed-loop poles originate from the open-loop poles
and terminate at the open loop zeros. If there is less zeros than poles then the some root loci originate
at open-loop poles
and increase towards infinity as K increases towards infinity.

The system has the best stability point at K = -1, at values below this root loci moves
towards the instability boundary.

Rules for Constructing Root Loci

These rules are listed with minimum clarification..For more details refer to reference
links and reference texts..
The rules below are simple rules which obviate the need to completely solve the characteristic
equation allowing the methods to be used for relatively complex systems. The rules are based
on those devised by R.Evans in an important paper in 1948. They are
therefore known as Evans Rules. The rules only relate to positive changes in K.
For negative values of K a set of similar rules are used.

The closed loop poles are roots of the characteristic equation for the closed loop system =

P(s).V(s) + K.Q(s)W(s) = 0

This can be more clearly expressed as

Poles(s) + K Zeros(s) = 0

1) Number of root loci.(branches)
The number of root loci is equal to the order of the characteristic equation .
This is, for rational systems, the same order as the characteristic
equation for the open loop transfer function i.e. P(s).V(s)

2) Symmetry of loci
The roots of the characteristic equation having real coefficient are symmetrical with respect
to the real axis

3) Poles
The poles of lie on the root loci and correspond to K = 0. i.e roots of P(s).V(s) = 0

4) Zeros
The zeros lie on the root loci and correspond to K = infinity. i.e roots of Q(s).W(s) = 0 .
If there are "t" more poles than zeros then "t" loci will become infinite as K approaches infinity

5) Asymptotes of root loci
If F(s) has "t" more poles than zeros, the root loci are asymptotic to "t" straight
lines making angles..

7) Root loci on real axis
If F(s) has one or more real poles or zeros ,then the segment of the real axis having
an odd number of real poles and zeros to its right will be occupied by a root locus.

8) Singularity/Breakaway Points
Singular points ( αb ) indicate the presence of multiple characteristic roots, and occur at
those values of s which for which dK/ds = 0..These points can also be obtained by solving the equation.
(- pi and - zi are the poles and zeros respectively)

(-pi and -zi are the poles and zeros respectively)

Singularity /breakaway points ( αb ) occur where two or more branches
of the root-locus depart from or arrive at the real axis.

9) Intersection of root loci with imaginary axis
The intersections of root loci with the imaginary axis can be located by calculating the
values of K which result in the imaginary characteristic roots.

10) Slopes of root loci at complex poles and zeros
The slope of a root loci at a complex pole or zero of F(s) can be found at at point in the
neighborhood of the pole or zero using the method shown below

11) Calculation of K on the root loci.
The absolute magnitude of the value of K corresponding to any point so on a root locus
can be found by measuring the lengths of the vectors drawn to so from the poles and zeros of F(s) and
then evaluating